Bunuel wrote:

Amy deposited $1,000 into an account that earns 8% annual interest compounded every 6 months.

Bob deposited $1,000 into an account that earns 8% annual interest compounded quarterly.

If neither Amy nor Bob makes any additional deposits or withdrawals, in 6 months how much more money will Bob have in his account than will Amy have in hers?

(A) $40

(B) $8

(C) $4

(D) $0.40

(E) $0.04

Amy - interest onlyEvery 6 months, Amy's interest is paid at the annual 8% rate divided by 2 (rate/# of compounding periods).

Every six months she earns \(\frac{.08}{2}=.04\) on her account balance (principal + any accrued interest)

At the 6-month mark, Amy gets her first interest payment of 4% (on $1,000).

Amy's INTEREST EARNED at 6 months: \((.04 * $1,000)= $40\)

Bob - interest onlyEvery 3 months, Bob's interest is paid at the annual 8% rate divided by 4 (rate/# of compounding periods).

Every 3 months, he earns \(\frac{.08}{4}=.02\) interest on his accumulated balance

1) After 3 months he gets paid \((.02 * $1,000) = $20\)

Now Bob has \($1,020\)

2) At the 6-month mark, he gets paid \((.02 * $1,020) = $20.40\)

Bob's INTEREST EARNED after six months: \(($20 + $20.40) = $40.40\)

Difference between Amy and Bob?

\($40.40 - $40.00 = $0.40\)

Answer D

Compound interest formula: \(A= P(1+\frac{r}{n})^{nt}\)

A = total amount, P = principal, r = annual interest rate in decimal form, n = number of interest payments in a year, and t = time in years

Amy - 8% annual compounded every 6 months

In half a year: \(A= $1,000(1+\frac{.08}{2})^{(2*\frac{1}{2})}\)

\(A=$1,000(1.04)^1=($1,000*1.04) = $1,040\)After six months, Amy's total is \($1,040\)

Bob - 8% annual compounded quarterly

In half a year, i.e., two quarters \(A=$1,000(1+\frac{.08}{4})^{(4*\frac{1}{2})}\)

\(A=$1,000(1.02)^2\)

\(1.02*1.02=1.0404\)

After 6 months Bob has \((1.0404 * $1,000)=$1,040.40\)

Difference? \($1,040.40 - $1,040.00 = $0.40\)

Answer D

_________________

The only thing more dangerous than ignorance is arrogance.

-- Albert Einstein